ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C1 VECTOR FIELDS

Title & Authors
ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C1 VECTOR FIELDS
Lee, Keonhee; Le, Huy Tien; Wen, Xiao;

Abstract
Let $\small{{\gamma}}$ be a hyperbolic closed orbit of a $\small{C^1}$ vector field X on a compact boundaryless Riemannian manifold M, and let $\small{C_X({\gamma})}$ be the chain component of X which contains $\small{{\gamma}}$. We say that $\small{C_X({\gamma})}$ is $\small{C^1}$ robustly shadowable if there is a $\small{C^1}$ neighborhood $\small{\mathcal{U}}$ of X such that for any $\small{Y{\in}\mathcal{U}}$, $\small{C_Y({\gamma}_Y)}$ is shadowable for $\small{Y_t}$, where $\small{{\gamma}_Y}$ denotes the continuation of $\small{{\gamma}}$ with respect to Y. In this paper, we prove that any $\small{C^1}$ robustly shadowable chain component $\small{C_X({\gamma})}$ does not contain a hyperbolic singularity, and it is hyperbolic if $\small{C_X({\gamma})}$ has no non-hyperbolic singularity.
Keywords
chain component;dominated splitting;homoclinic class;hyperbolicity;robust shadowability;uniform hyperbolicity;vector field;
Language
English
Cited by
1.

대한수학회보, 2015. vol.52. 1, pp.149-157
1.
SHADOWABLE CHAIN COMPONENTS AND HYPERBOLICITY, Bulletin of the Korean Mathematical Society, 2015, 52, 1, 149
2.
Weak measure expansive flows, Journal of Differential Equations, 2016, 260, 2, 1078
3.
HYPERBOLICITY OF HOMOCLINIC CLASSES OF VECTOR FIELDS, Journal of the Australian Mathematical Society, 2015, 98, 03, 375
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